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This page should include several computations of sheaf cohomology. This should include
cohomology of projective space
cohomology of the module from the structure sheaf of a projective variety
hodge theory computations
This page should also describe applications. One is the fact that the sheaf cohomology modules can be used to construct examples of families of extensions parameterized by the affine line using the fact that we can just a morphism of into one of these vector spaces (since we can use the derived category interpretation). This shows that is not invariant since if our map contains and a non-zero point, we have a trivial and non-trivial extension of vector bundles, hence they are not isomorphic. — Preceding
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01:46, 10 October 2017 (UTC)reply
Here are some additional resources for computing cohomology:
For example, take a complete intersection surface of bidegree in . Then, using the Hodge theory notes there are short exact sequences on the degree hypersurface from the cotangent sequence
Then, using the cotangent sequence again we find
These give long exact sequences, which can then be used to compute the hodge numbers of this complete intersection hypersurface. The other two sequences to consider are
Notice that tensoring the euler sequence will show how to compute the cohomology of the first sequence.